Interesting comparisons. They do a great job of showing how the N64's more feature-rich polygons allow developers to use fewer of them since they don't need to cover over visible polygon seams.
The PS1 did on average push more polygons than the N64, and few N64 games match the polygon counts of the top PS1 games, but the main reason for the huge disparity you see in those wireframes is of course that the PS1 doesn't have perspective correction, Z-buffering, etc., so in order to cover up the visible polygon seams developers were forced to use many small overlapping polygons.
On the N64, of course, large polygons display just fine with no edge problems, so you can use many fewer polygons as a result. The high-poly flat surfaces you see in those pictures are entirely a result of trying to cover up the PS1's low-accuracy 3d. It helps, but does not entirely get rid of the problem; all PS1 games have polygon-seam issues somewhere or other.
I'm sure the PS1 can do big polygons too, but large, non-overlapped polygons would have horrendous poly-seam issues! It looks terrible, which is why eventually developers started doing what you see in those pictures there.